Let G be a group of order
Then, G is a finite group and the prime p divides the order of G, i.e.,
Proving the result by induction on n,
Thus, the group has only one element of order 1. The result holds for n=1
Next, suppose every group with pk elements contains an element of order p for every k Consider the class equation of G,
If G=Z(G), then G is abelian and hence the result follows.
Then, for every a !in Z(G),
If for some
Otherwise, for every
Thus, it follows that Z(G) and hence G contains an element of order p.